## Heat Transfer from a Fin **Maximum possible heat transfer** (if the entire fin were at base temperature): $Q_{\text{max}} = h A_f (T_b - T_\infty)$ **Actual heat transfer** depends on geometry and tip condition. For example, for an adiabatic tip: $Q_f = \sqrt{hPkA_c} (T_b - T_\infty) \tanh(mL)$ **Fin efficiency** (how well a fin performs compared to ideal): $\eta_f = \frac{Q_{\text{actual}}}{Q_{\text{max}}}$ ## Fin Resistance - Fin resistance (includes efficiency): $R_f = \frac{1}{\eta_f h A_f}$ ## Fin Parameter and Definitions Fin parameter: $m = \sqrt{\frac{hP}{kA_c}}$ - Fin surface area: $A_f$ Fin perimeter: $P$ Cross-sectional area: $A_c$ - Modified fin heat transfer term: $M = h P k A_c (T_b - T_\infty)$ ## Efficiency Expressions (Special Cases) - Infinite fin: $\eta_f = \frac{1}{mL}$ - Adiabatic tip: $\eta_f = \frac{\tanh(mL)}{mL}$ ## Governing Equation and Solution - 1D conduction + convection (steady state): $\frac{d^2 \theta}{dx^2} - m^2 \theta = 0$, where $\theta(x) = T(x) - T_\infty$ - General solution: $\theta(x) = A e^{mx} + B e^{-mx}$ - With adiabatic tip boundary condition: $\frac{d\theta}{dx}\big|_{x=L} = 0$ - With prescribed tip temperature: $\theta(L) = \theta_L$ ## Heat Transfer Rate (Adiabatic Tip) - Heat conducted at base: $Q_f = \sqrt{hPkA_c} (T_b - T_\infty) \tanh(mL)$ - Can also be written: $Q_f = M \tanh(mL)$ ### Fin Tip Conditions | Tip Condition | Temperature Distribution $\theta(x)/\theta_b$ | Fin Heat Transfer Rate $Q_f$ | |------------------------|---------------------------------------------------------------------------|--------------------------------------------| | Infinite fin | $\theta(x)/\theta_b = e^{-mx}$ | $Q_f = M$ | | Adiabatic tip | $\theta(x)/\theta_b = \frac{\cosh(m(L-x))}{\cosh(mL)}$ | $Q_f = M \tanh(mL)$ | | Prescribed temp at tip | $\theta(x)/\theta_b = \frac{\sinh(mL) + \sinh(m(L-x))}{\sinh(mL)}$ | $Q_f = M \left(\frac{\cosh(mL) - 1}{\sinh(mL)}\right)$ | Where: - $\theta(x) = T(x) - T_\infty$ - $\theta_b = T_b - T_\infty$ - $m = \sqrt{\frac{hP}{kA_c}}$ - $M = \sqrt{hPkA_c} (T_b - T_\infty)$